# Asymptotic behavior of eigenvalues and eigenvectors of a random matrix

## Abstract/Contents

- Abstract
- This thesis can be mainly divided into two parts. In the first part, we consider an N-by-p data matrix X_N where all the rows are i.i.d. p-dimensional samples of mean zero and covariance matrix Sigma. Here the population matrix Sigma is of finite rank perturbation of the identity matrix. This is the ``spiked population model'' first proposed by Johnstone. As N, p both go to infinity but N/p tends to a fixed ratio, for the sample covariance matrix, we establish the joint distribution of the largest and the smallest few packs of eigenvalues. Inside each pack, they will behave the same as the eigenvalues drawn from a Gaussian matrix of the corresponding size. Among different packs, we also calculate the covariance between the Gaussian matrices entries. As a corollary, if all the rows of the data matrix are Gaussian, then these packs will be asymptotically independent. Also, the asymptotic behavior of sample eigenvectors are obtained. Their local fluctuation is also Gaussian with covariance explicitly calculated. As the second part of the thesis, we study the limiting distribution of the k smallest gaps between eigenvalues of three kinds of random matrices -- the Ginibre ensemble, the Wishart ensemble and the universal unitary ensemble. All of them follow a Poissonian ansatz. More precisely, for the Ginibre ensemble we have a global result in which the k-th smallest gap has typical length n^{-3/4} with density x^{4k-1}e^{-x^4} after normalization. For the Wishart and the universal unitary ensemble, it has typical length n^{-4/3} and has density x^{3k-1}e^{-x^3} after normalization.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2013 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Shi, Dai | |
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Associated with | Stanford University, Institute for Computational and Mathematical Engineering. | |

Primary advisor | Papanicolaou, George | |

Thesis advisor | Papanicolaou, George | |

Thesis advisor | Dembo, Amir | |

Thesis advisor | Johnstone, Iain | |

Advisor | Dembo, Amir | |

Advisor | Johnstone, Iain |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Dai Shi. |
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Note | Submitted to the Institute for Computational and Mathematical Engineering. |

Thesis | Thesis (Ph.D.)--Stanford University, 2013. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2013 by Dai Shi
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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